Traces for fractional Sobolev spaces with variable exponents

Abstract

In this note we prove a trace theorem in fractional spaces with variable exponents. To be more precise, we show that if $p\colon\overlineΩ\times \overlineΩ\to (1,\infty)$ and $q:\partial Ω\rightarrow (1,\infty)$ are continuous functions such that \[ \frac{(n-1)p(x,x)}{n-sp(x,x)}>q(x) \qquad \mbox{in} \partial Ω\cap \{x\in\overlineΩ\colon n-sp(x,x) >0\}, \] then the inequality $$ \Vert f\Vert _{\scriptstyle L^{q(\cdot)}(\partial Ω)} \leq C \left\{\Vert f\Vert _{\scriptstyle L^{\bar{p}(\cdot)}(Ω)}+ [f]_{s,p(\cdot,\cdot)} \right\} $$ holds. Here $\bar{p}(x)=p(x,x)$ and $\lbrack f\rbrack_{s,p(\cdot,\cdot)} $ denotes the fractional seminorm with variable exponent, that is given by \[ \lbrack f\rbrack_{s,p(\cdot,\cdot)} := \inf \left\{λ>0\colon \int_Ω\int_Ω\frac{|f(x)-f(y)|^{p(x,y)}}{λ^{p(x,y)} |x-y|^{n+sp(x,y)}}dxdy<1\right\} \] and $\Vert f\Vert _{\scriptstyle L^{q(\cdot)}(\partial Ω)}$ and $\Vert f\Vert _{\scriptstyle L^{\bar{p}(\cdot)}(Ω)}$ are the usual Lebesgue norms with variable exponent.